CoE Advising Report

116 Pages·2010·6.65 MB·English

Chapter 13
The Pros and Cons of Diffusion, Filters and
Fixers in Atmospheric General Circulation
Models
ChristianeJablonowskiandDavidL.Williamson
AbstractAllatmosphericGeneralCirculationModels(GCMs)needsomeformof
dissipation,eitherexplicitlyspecifiedorinherentinthechosennumericalschemes
forthe spatial andtemporaldiscretizations.Thisdissipationmayserve manypur-
poses,includingcleaningupnumericalnoisegeneratedbydispersionerrorsorcom-
putational modes, and the Gibbs ringing in spectral models. Damping processes
mightalsobeusedtocrudelyrepresentsubgridReynoldsstresses,eliminateunde-
sirable noise due to poor initialization or grid-scale forcing from the physics pa-
rameterizations,coverup weakcomputationalstability, damptracervariance,and
preventtheaccumulationofpotentialenstrophyorenergyatthesmallestgridscales.
ThischaptercriticallyreviewsthewideselectionofdissipativeprocessesinGCMs.
Theyaretheexplicitlyaddeddiffusionandhyper-diffusionmechanisms,divergence
damping, vorticity damping, external mode damping, sponge layers, spatial and
temporalfilters,inherentdiffusionpropertiesofthenumericalschemes,andaposte-
riorifixersusedtorestorelostconservationproperties.Alltheoreticalconsiderations
aresupportedbymanypracticalexamplesfromawideselectionofGCMs.Theex-
amplesutilizeidealizedtestcasestoisolatecausesandeffects,andtherebyhighlight
theprosandconsofthediffusion,filtersandfixersinGCMs.
13.1 Introduction
Therearemanydesignaspectsthat needto beconsideredwhenbuildingthe fluid
dynamicscomponent,theso-calleddynamicalcore,foratmosphericGeneralCircu-
ChristianeJablonowski
DepartmentofAtmospheric,OceanicandSpaceSciences,UniversityofMichigan,2455Hayward
St.,AnnArbor,MI48109,USA,e-mail:[email protected]
DavidL.Williamson
NationalCenterforAtmosphericResearch, 1850TableMesaDrive,Boulder,CO80305,USA,
e-mail:[email protected]
387
388 ChristianeJablonowskiandDavidL.Williamson
lation Models(GCMs). Amongthem arethe choice ofthe equationset andprog-
nosticvariables,thecomputationalgridandgridstaggeringoptions,thespatialand
temporalnumericaldiscretizations,built-inconservationproperties,andthechoice
of dissipative processes that (1) might be needed to keep a model simulation nu-
mericallystableand(2)mighttruthfullymimicthecumulativeeffectsofunresolved
subgrid-scale processes on the resolved fluid flow. The latter aspect is at least a
“hope”in theGCM modelingcommunity.Here,thephrasesubgrid-scaledenotes
thedryadiabaticunresolvedprocessesinthedynamicalcore.Thisisincontrastto
allotherunresolvedprocessesthatleadtophysicalparameterizationssuchasradi-
ation,convection,cloudprocessesandplanetaryboundarylayerphenomena.These
arenotconsideredhere,eventhoughtheyareintimatelycoupledtotheequationsof
motion.Thischaptershedslightontheprosandconsofthemostpopularprocesses
to handle both “physical”or “unphysical”subgrid-scale flow and mixing,and re-
viewstheuseofexplicitlyaddedandinherentdiffusion,filtersandfixersinGCMs.
ThesearerarelydocumentedintherefereedGCMliteraturebutmightbedetailed
intechnicalmodeldescriptions.
It is commonpractice in GCMs to include a parameterizationof the effects of
subgrid-scalemotionsin the horizontalmomentumand thermodynamicequations
that is formulated as a local diffusive mixing. In fact, all numerical models need
some form of dissipation, either explicitly specified or inherentin the chosen nu-
mericalschemesforthespatialandtemporaldiscretizations.Thisdissipationmay
serve manypurposes,includingcleaningup numericalnoisegeneratedby disper-
sion errors,computationalmodes,or the Gibbsringing,crudelyrepresentingsub-
grid Reynoldsstresses, eliminating undesirable noise dueto poor initialization or
grid-scaleforcingfromthephysicsparameterizations,coveringupweakcomputa-
tionalstability,dampingtracervariance,andpreventingtheaccumulationofpoten-
tialenstrophyorenergyatthegridscale(Woodetal2007;Thuburn2008a).Such
an accumulation of energy is due to the physical downscale cascade and can re-
sult in excessive small-scale noise. It is known as spectral blocking and leads to
anupturn(hook)orflatteninginthekineticenergyspectrumatthesmallestscales.
Furthermore,physical“noise”in GCMs mightoriginatefromparameterizedgrid-
scaleforcingsorfromsurfaceboundaryconditionssuchasorography,theland-sea
orland-usemask.
Anaccumulationofenergyandenstrophyatthe smallest scalesmayalsoarise
duetoanumericalmisrepresentationofnonlinearinteractions,theso-calledalias-
ingeffect.Nonlinearinteractionsandaliasingmostlyoriginatefromthe quadratic
orhigher-ordertermsintheequationsofmotion.Inessence,productsofwavescan
create new waves that are shorter than 2!x where !x is the physical grid spac-
ing.Thesewavescannotberepresentedonamodelgridandarealiasedintolonger
waves.Aliasing,ifleftunchecked,canleadtoablowupofthesolution.Thisphe-
nomenonis characterizedas nonlinear computationalinstability as first discussed
by Phillips (1959). Note that almost all GCMs suffer to some degree from alias-
ing.Exceptionsarespectraltransformmodelswithquadratictransformgridswhich
eliminatethealiasingofquadraticadvectionterms,themostproblematicform,but
donotcompletelyeliminatealiasingfromhigher-orderterms.Nonlinearcomputa-
13 Diffusion,filtersandfixersinGCMs 389
tional instability does not occur in models that conserve quadratic quantities like
enstrophyandkineticenergy(Arakawa1966;ArakawaandLamb1981).Aliasing
errorsare not necessarily fatal. Whether an amplification of the waves occursde-
pendsonthe phaserelationbetweenthemisrepresentedandoriginalwavesinthe
model.Moreinformationonnonlinearcomputationalinstabilityandaliasingispro-
videdintextbookslikeDurran(1999,2010),Kalnay(2003)orLin(2007).
All mixingprocessesremoveenergyand enstrophyfrom the simulation which
would otherwise build up to unrealistic proportions.Frequently, the included dis-
sipation is restricted to be in the horizontal, as there is usually sufficient vertical
mixing or diffusion in the physical boundarylayer turbulence and convectivepa-
rameterizationsin full GCMs or sufficient inherentnumerical diffusionto control
noise in the vertical direction. Sometimes, vertical diffusion is also explicitly in-
cluded in the dynamical core and applied throughout the whole troposphere. An
exampleisthemodelbyTomitaandSatoh(2004)whichisdiscussedlater.
A common expectation might be that dissipative formulationsbased on turbu-
lence theory or observations provide a physical foundation for the subgrid-scale
mixing.However,suchphysicalmotivationisnotguaranteedandeachadhocmix-
ingprocessinaGCMmustbecriticallyreviewed.AspointedoutbyMellor(1985)
the horizontal diffusivities in use by GCMs are typically many orders of magni-
tude larger than those which would be appropriate for turbulence closures. Thus,
horizontaldiffusionusedbymostmodelscannotbeconsideredarepresentationof
turbulencebutshouldbeviewedasasubstitutemechanismforunresolvedhorizon-
tal advectiveprocesses.Awarenessof thismightoffersomeguidancein choosing
anadequatesubgrid-scalemixingscheme.
Mixing in GCMs generally serves as a numerical filter and neither reflects the
mathematical representation of the energy or enstrophy transfer to small scales
nor the representation of physical molecular diffusion (Koshyk and Boer 1995).
Subgrid-scaleprocesses,althoughsmall,canhaveaprofoundimpactonthelarge-
scale circulation. For example, diffusive mechanisms affect the propagation of
wavesandtherebythemeanflowthroughwave-meanflowinteractions.Inaddition,
bothinherentandexplicitlyaddeddissipationprocessessmearoutsharpgradients
inthetracerfields,andmayleadtounphysicalandoverlystrongmixing.Suchmix-
ing might provide feedbacks to the physical parameterizations. For example, the
precipitationfieldmightbehighlyinfluencedbythediffusivecharacteristicsofthe
moisturetransportalgorithminthedynamicalcore.Thenotionofoverlydiffusive
GCMswasdiscussedbyShutts(2005).Hearguedthatnumericaladvectionerrors,
horizontal diffusion and parameterization schemes like the gravity wave drag or
convection,actasenergysinksandleadtoexcessiveenergydissipationinGCMs.
However,suchaconclusionmightbehighlymodeldependent.
In summary, some mixing processes are used for purely numerical reasons to
prevent the model from becoming unstable. Others are meant to mimic subgrid-
scaleturbulenceprocessesthatareunsolvedonthechosenmodelgrid.Inpractice,
manyfiltersandmixingprocessesareusedatonce,whichmakesitmoredifficultto
evaluatetheirindividualeffects.Theformofthediffusionprocessesinatmospheric
dynamical cores varies widely and is somewhat arbitrary. There are explicit dis-
390 ChristianeJablonowskiandDavidL.Williamson
sipationprocessesandfilters,inherentnumericaldissipation,andfixersin GCMs.
Throughoutthechapterweassociatethephraseexplicitdiffusionwithprocessesex-
plicityaddedtotheequationsofmotion.Thephraseimplicitdiffusioncharacterizes
theinherentdissipationofnumericalschemes.Thesephrasesareintendedtomake
adistinctionfromexplicitandimplicitnumericalapproximationstodiffusionoper-
ators.Notethatthewords“diffusion”,“dissipation”and“viscosity”areoftenused
interchangeablyintheliterature.Othercharacterizationsofdampingaresmoothing,
filteringandmixing.
13.1.1 Modelequationsandtherepresentationofexplicitdiffusion
Mixing processes in GCMs can appear in many forms. A very dominant form is
based on explicit dissipation mechanisms that get appended to the equations of
motion shown in chapter 15. In the continuous equations this mixing symbolizes
moleculardiffusion.However,GCMsarenotcapableofrepresentingmoleculardif-
fusionat thenmormmscalesincetheyaretypicallyappliedwithhorizontalgrid
spacingsbetween20-300km.NonhydrostaticGCMs(Tomitaetal2005;Fudeyasu
etal2008)andmesoscalelimited-areamodelsliketheWeatherResearchandFore-
casting Model WRF (Skamarocket al 2008)are also run with finer grid spacings
ofafewkilometers.Otheratmosphericmodelswithevenfinerscalesmightutilize
theLargeEddySimulation(LES)approach.LESisamathematicalmodelfortur-
bulencethatisbasedupontheNavier-Stokesequationswithbuilt-inlow-passfilter.
TheunderlyingideawasinitiallyproposedbySmagorinsky(1963)andfurtherde-
velopedby Deardorff(1970).LES hasbeenextensivelyused to studysmall-scale
physicalprocessesandmixingintheatmosphere.Butinanycase,modelstruncate
themulti-scalespectrumofatmosphericmotionswellabovethemoleculardiffusion
scales.Theunresolvedpartistypicallymodeledasdissipationandonemighthope
thatitadequatelycapturestheadiabaticsubgrid-scaleprocessesinsomepoorlyun-
derstoodway.
Explicit dissipation can be added to the momentumand thermodynamicequa-
tionsinthesymbolicform
"#
=Dyn(#)+Phys(#)+F (13.1)
#
"t
whereDyn(#)denotesthetimetendenciesoftheprognosticvariable#according
totheresolvedadiabaticdynamics,Phys(#)symbolizesthetime tendenciesfrom
thesubgrid-scalediabaticphysicalparameterizations,andF isthedissipation.The
#
actualformofthisdissipationismodeldependent.Forexample,modelsinmomen-
tum form,that utilize the zonal and meridionalvelocitiesu, v and temperatureT,
might append the diffusive terms Fu,Fv,FT. Models in vorticity-divergence($,%)
formaddthediffusionF ,F ,F ,orevenreplaceF withadiffusionoftheabsolute
$ % T $
vorticityF where f symbolizestheCoriolisparameter.Alternatively,ifthepo-
$+f
tentialtemperature&isselectedinthethermodynamicequationadiffusivetermF
&
13 Diffusion,filtersandfixersinGCMs 391
mightbechosen.Dissipationmightalsobeappliedtothetracertransportequations,
andincaseofnonhydrostaticmodelstotheverticalvelocity.Whetherexplicitdiffu-
sionisneededforcomputationalstabilityismodeldependent.Somemodelsprefer
tocontrolthesmallestscalesviainherentnumericaldissipationandselectF =0.
#
However,theformofF isoneofthemainfociinsections13.3-13.5,andtherefore
#
weintroducethegenericformoftheforcinghere.
13.1.2 Overviewofthechapter
Thischapterpresentsacomprehensivereviewofdissipativeprocessesandfixersin
generalcirculationmodels.Manypointerstoreferencesaregiven,andweillustrate
thepracticalimplicationsofthediffusion,filtersandfixersonthefluidflowinatmo-
sphericdynamicalcores.Inparticular,wereviewtheprinciplesbehindthedifferent
dissipative formulations, isolate causes and effects, provide many examples from
today’sGCMs and utilize idealizeddynamicalcoretest cases andso-calledaqua-
planetsimulationstodemonstratetheconcepts.Thesetestcasesarebrieflyoutlined
insection13.2.Overall,wequoteorshowexamplesfromover20differentdynami-
calcorestohighlightthebroadspectrumofthedissipativeprocessesinGCMs.The
modelsarelistedinsection13.2.Wecharacterizefourteenofthemingreaterdetail
intheAppendixsincetheyareusedasexamplesthroughoutthechapter.
Thechapterisorganizedasfollows.Sections13.3and13.4discussthemostpop-
ularexplicitdiffusionanddampingmechanismsinthedynamicalcoresofGCMs.
Section 13.3 includes the classical linear and nonlinear horizontal diffusion and
hyper-diffusion,their diffusion coefficients and stability constraints, and physical
consistency arguments. Section 13.4 discusses the 2D and 3D divergence damp-
ing,vorticitydamping,Rayleighfrictionanddiffusivespongesnearthemodeltop,
and externalmode damping.In general,it is debatablewhether filters are consid-
eredexplicitdissipationorjustacomputationaltechniquetokeepamodelnumeri-
callystable.Here,wechoosetopresentthemintheirowncategoryinsection13.5
wherebothtemporalandspatialfilterareassessed.Section13.6capturesthebasic
ideas behind inherent numerical dissipation which is nonlinear and sometimes is
interpretedasphysicallymotivateddiffusion.Section13.7shedslightonthecon-
servationpropertiesofatmosphericGCMsandintroducesaposteriorifixers.They
includethedryairmassfixer,fixersfortracermassesandtotalenergyfixers.Some
finalthoughtsarepresentedinsection13.8.
13.2 SelectedDynamicalCoresandTest Cases
We illustrate manyof the effectsof the diffusion,filters and fixersin GCMs with
thehelpof2Dshallowwateror3Dhydrostaticmodelrunstodiscussthepractical
implications of the theoretical concepts. Throughoutthis chapter, we point to the
392 ChristianeJablonowskiandDavidL.Williamson
specificimplementationsofthedissipativeprocesses,andquotetypicalvaluesfor
the empirical coefficients from a variety of models to shows their spread in the
GCMs.Theintentionistogivehands-onguidanceandpresentthedesignoptions.
Inparticular,thischapterfeaturesexamplesfromthedynamicalcoresCAMEu-
lerian, CAM semi-Lagrangian,COSMO, ECHAM5, FV, FVcubed, GEOS, GME,
HOMME,ICON,IFS,NICAM,UMandWRF.TheAppendixexplainstheacronyms
andbrieflycharacterizesthenumericalschemes.Mostmodelsareglobalhydrostatic
GCMsandusetheshallow-atmosphereapproximation.Theonlyexceptionsarethe
nonhydrostatic models COSMO, NICAM, UM and WRF that are built upon the
deepatmosphereequationset(seeWhiteetal(2005)forareviewoftheequations).
WealsobrieflyrefertoothermodelssuchasNASA’sModelEbytheGoddardInsti-
tuteforSpaceStudies,theAtmosphericGCMfortheEarthSimulator(AFES)de-
velopedbytheCenterforClimateSystemResearchattheUniversityofTokyoand
the National Institute for EnvironmentalStudies (Japan),the Global Environmen-
talMultiscale(GEM)modelfromtheCanadianMeteorologicalCentre,theGlobal
Forecast System (GFS) and the Eta model developedby the National Centers for
EnvironmentalPrediction(NCEP).Thereferencesforthesemodelsaregivenlater.
The model simulations utilize a variety of idealized test cases. They include
the steady-state and baroclinic wave test cases for dynamical cores suggested by
JablonowskiandWilliamson(2006a,b),selectedshallowwatertestcasesfromthe
Williamsonetal(1992)testsuite,theHeld-Suarezclimateforcing(HeldandSuarez
1994), a variant of the Held-Suarez test with modified equilibrium temperatures
in the stratosphere (Williamson et al 1998), and the aqua-planet configuration as
proposedbyNealeandHoskins(2000).Theadiabaticdynamicalcoreandshallow
watertestcasesgenerallyassessthepropertiesofthenumericalschemesinshortde-
terministicmodelrunsofupto30days.TheidealizedHeld-Suarez-typesimulations
utilize a prescribedNewtoniantemperaturerelaxationandboundarylayerfriction
thatreplacethephysicalparameterizationpackage.Thesemodelrunsaretypically
integratedformultipleyearstoassess theclimatestatistics ona dryandspherical
earthwithnomountains.Theaqua-planetassessmentsarethemostcomplexsimula-
tionsdiscussedinthischapter.TheyrepresentmoistGCMrunsthatincludethefull
physicalparameterizationsuitebututilizeasimplifiedlowerboundarycondition.In
essence, the lower boundaryis replaced by a water coveredearth with prescribed
sea surface temperatures. In addition, the settings of the physical constants and a
symmetricozonedatasetareprescribedinaqua-planetsimulations.
13.3 ExplicitHorizontalDiffusion
Thissectiondiscussestheideasbehindexplicithorizontaldiffusionmechanismsin
GCMs. Inparticular,we assess thelinearsecond-orderdiffusion,the higher-order
and thereby more scale-selective hyper-diffusion, reveal the selection criteria for
thediffusioncoefficientsinbothspectraltransformandgridpointmodels,discuss
theconceptofspectral viscosity,andreviewthe stabilityconstraintsforthediffu-
13 Diffusion,filtersandfixersinGCMs 393
sionequation.Inaddition,weintroducetheprinciplesbehindnonlinearhorizontal
diffusionandbrieflysurveythephysicalconsistencyofexplicitdiffusionschemes.
13.3.1 Genericformoftheexplicitdiffusionmechanism
Thegenericformoftheexplicitlineardiffusionisgivenby
F#=( 1)q+1K2q’2q# (13.2)
−
whereq 1isapositiveinteger,2qdenotestheorderofthediffusion,K stands
2q
≥
forthediffusioncoefficientand’isthegradientoperator.Boththehorizontal2D
gradientoperatororanapproximated3Dgradientoperatorhavebeenusedforthe
horizontaldiffusionasfurtherexplainedbelow.Settingq=1yieldsasecond-order
diffusionthatemergesfromphysicalprinciplessuchastheheatdiffusion,molecular
diffusionandBrownianmotion.However,moleculardiffusionactsonthenanome-
tertomillimeterscale,andisthereforeunresolvedonaGCMmodelgrid.
In practice, second-orderdiffusionis often applied as an artificial sponge near
the topboundary,andhasverylittle resemblancewith itsphysicalcounterpart.In
general,morescale-selectivehyper-diffusionschemeswithq=2,3,4areselected
in the majorityof the modeldomain.The most popularchoiceis the fourth-order
hyper-diffusionwithq=2thatisalsocalledbi-harmonicdiffusionorsuperviscos-
ity.Theuseofhyper-diffusionisoftenmotivatedbytheneedtomaximizetheratio
of enstrophy to energy dissipation since 2D turbulence theory predicts a vanish-
ingenergydissipationrateatincreasingReynoldsnumbers(SadournyandMaynard
1997).Thehighertheorderofthehyper-diffusion,thehighertheratioofenstrophy
toenergydissipationbecomes.FargeandSadourny(1989)evensuggestedusinga
16th-orderhyper-diffusion.
13.3.2 ParticularformsofexplicitdiffusioninGCMs
TheexactformofthedampingvarieswidelyinGCMs.Typically,forconvenience
thehorizontaloperatorsareappliedalongmodellevelswiththepossibleexception
oftheformulationforthescalartemperaturediffusion.Wenowlistseveralexamples
toillustratethevarietyofthediffusionmechanisms.Ourfirstexampleistakenfrom
theweatherforecastmodelGMEwhichhasbeendevelopedattheGermanWeather
Service.Itappliesthefourth-orderhyper-diffusion
Fu = K4’4u (13.3)
−
Fv = K4’4v (13.4)
−
FT = K4’4(T Tref) (13.5)
− −
394 ChristianeJablonowskiandDavidL.Williamson
where’isthehorizontalgradientoperatorandTref isareferencetemperaturethat
dependsonlyonheight(Majewskietal2002).Atupperlevelsnearthemodeltop,
asecond-orderdiffusionisapplied.GMEutilizeslocalbasisfunctionsthatareor-
thogonalandconformperfectlytothesphericalsurface.Theyarelocallyanchored
ineachtriangleofGME’sicosahedralgrid.Withinthelocalneighborhoodofagrid
point the coordinate system is thereforenearly Cartesian.Note that Cartesian co-
ordinatessimplifytherepresentationofthe’2q operatorsincethemetrictermsare
equaltounity.
If the diffusion is expressed in spherical coordinates many metric terms are
present.Here,wefirstshowtheoperatorsforthreespatialdimensionsbeforesim-
plifyingthem.Thescalar3DLaplacian’2 operatorinsphericalcoordinatesfor
(3D)
aprognosticvariable#hastheform
1 1 1
’2(3D)#= r2cos2("))#+r2cos("((cos("(#)+r2"r(r2"r#) (13.6)
whererdenotestheradialdistanceinthelocalverticaldirectionfromthecenterof
theearth,)and(arethelongitudeandlatitude,andthenotation"x symbolizesa
partialderivativeinthexdirectionwherexisaplaceholderfor),(andr.Inaddi-
tion,the3DvectorLaplacianforthethree-dimensionalwindvectorv =(u,v,w)is
3
givenby
2sin( 2 1
’2 u " v+ " w u
(3D) −r2cos2( ) r2cos( ) −r2cos2(
 1 2 2sin( 
’2 v = ’2 v v+ " w+ " u . (13.7)
(3D) 3  (3D) −r2cos2( r2 ( r2cos2( ) 
 2 2 2 
’2 w w " u " (cos(v)
 (3D) −r2 −r2cos( ) −r2cos( ( 
 
 
Theseexpressionsaree.g.showninAppendixAinSatoh(2004).Theextraterms
besidesthescalar’2 operatorariseduetothespatialvariationoftheunitvectors
(3D)
insphericalcoordinates.Withtheexceptionoftheundifferentiatedtermineachof
the componentsthe extra terms are not necessarily negligiblein comparisonwith
thoseofthescalardiffusionoperator.Infact,someofthemarecrucialinensuring
thatthediffusionoperatorconservesangularmomentumasoutlinedbyStaniforth
etal(2006).
Generally, approximated forms of (13.6) and (13.7) are chosen to express the
horizontaldiffusion.Thedistancerisoftenapproximatedbytheconstantradiusof
theeartha,thetermscontainingtheverticalderivative"randverticalvelocityware
dropped,andtheverticalcomponentofthevectorLaplacianisneglectedtocreatea
2Ddiffusionoperator.Thereplacementofthedistancerbyaisinfactanecessityin
hydrostaticmodelsbasedontheprimitiveequations(Whiteetal2005).Thescalar
2DLaplacianthensimplifiesinthefollowingway
1 1
’2#= " #+ " (cos(" #). (13.8)
a2cos2( )) a2cos( ( (
13 Diffusion,filtersandfixersinGCMs 395
The2DvectorLaplacianforthetwo-dimensionalwindvectorv=(u,v)isgivenby
2sin( 1
’2u " v u
’2v= −a2cos2( ) −a2cos2( . (13.9)
 1 2sin( 
’2v v+ " u
−a2cos2( a2cos2( )
 
 
ThisformofthevectorLaplacianleadstothe“conventionalform”ofthehorizontal
momentum diffusion as characterized by Becker (2001). Unfortunately,this form
doesnotconserveangularmomentumasfurtherdiscussedinsection13.3.7.Some
modelsalsodroptheextratermsandonlyapplythescalar2DLaplacianoperatorto
thevectorwind(u,v)insphericalgeometry.Suchasimplifiedformise.g.provided
as an optionalspongelayer dampingmechanismnear the modeltop in the finite-
volume(FV)dynamicalcoreintheCommunityAtmosphereModelCAM(version
5)(Nealeetal2010).ThemodelCAMFVisusedattheNationalCenterforAtmo-
sphericResearch(NCAR).
There is another caveat. Both formulations of the Laplacian in Eqs. (13.7) or
(13.9)wouldleadto anundesireddampingofa solidbodyrotationasthoroughly
analyzed by Staniforthet al (2006)for the Unified Model (UM) developedat the
UKMetOffice.Thereforeinpractice,amorecomplicatedformofthemomentum
diffusionischoseninthemodelUMthatisappliedtothevelocitycomponentsu,v
and w (see Staniforth et al (2006) for the derivation). The NCAR CAM spectral
transform Eulerian (EUL) and semi-Lagrangian (SLD) dynamical cores (Collins
etal2004)alsoincludesuchacorrectionforsolidbodyrotationasexplainedlater.
Concerningthe scalar diffusionin the modelUM, a form similar to Eq. (13.8)
isselectedforthediffusionofpotentialtemperature.Itisappliedtwice(including
a sign reversal) to resemble a fourth-orderhyper-diffusionmechanism. The main
differences to Eq. (13.8) are that (1) the model UM does not utilize a shallow-
atmosphereapproximationand retainsthe radialdistancer, (2)a slope correction
is utilized over steep terrain to lessen the spurious mixing along UM’s deformed
orography-followingverticalcoordinate,and(3)thediffusioncoefficientisdiffer-
ent in the two horizontal directions. The coefficient is constant in the meridional
direction, but the strength of the diffusion in longitudinal direction is allowed to
varywithlatitude.Thisleadstonon-isotropicdiffusionandisfurtherexplainedin
section13.3.5.NotethatthemodelUMdoesnotneedhorizontaldiffusionforcom-
putationalstabilityreasonsduetotheinherentnumericaldissipationintheinterpo-
lationsofitssemi-Lagrangianscheme.Inpractice,theexplicitdiffusionistherefore
optionalandnotusedbydefault.Forexample,itisneverutilizedinshortweather
predictionsimulations(TerryDavies,personalcommunication).
Scalardiffusionoftype(13.8)isalsoappliedinothermodelssuchasthespectral
transformIntegratedForecastingSystem(IFS)attheEuropeanCentreforMedium-
RangeWeatherForecast(Ritchieetal1995;ECMWF2010).Themodelutilizesa
second-order(q=1)diffusionschemeclosetothemodeltopandafourth-order(q=
2)hyper-diffusionoftheprognosticscalarvariablesrelativevorticity$,horizontal
divergence%andtemperature.Ityieldstheexplicitdiffusion
396 ChristianeJablonowskiandDavidL.Williamson
F$ =(−1)q+1K2q’2q$ (13.10)
F% =(−1)q+1K2q’2q% (13.11)
FT =( 1)q+1K2q’2qT. (13.12)
−
Thisformofthe diffusionisfurthermoreutilizedbythespectraltransformmodel
ECHAM5 developed at the Max-Planck Institute for Meteorology, where even
higher-order diffusion operators are chosen below the sponge layer at the model
top(Roeckneretal2003).
Theapplicationofthediffusionalongslopinggeneralverticalcoordinates,like
thehybridpressure-based*-coordinate(SimmonsandBurridge1981),isstraight-
forward to implement, but as mentioned before can cause spurious mixing over
mountains, especially in the neighborhoodof steep terrain. This is largely due to
thepresenceoflargeverticaltemperaturevariationsalongtheslopingsurfacesthat
overlaythehorizontalgradients.Suchspuriousmixingtriggeredbytheverticalvari-
ationsisundesirableandmaygrowtosignificantproportions.Thereforeinpractice,
the diffusion of the temperature in the model IFS is modified to approximatethe
horizontaldiffusiononsurfacesofconstantpressureratherthanonthesloping*-
coordinatesurfaces.Thisisfurtherexplainedinthetechnicalmodeldocumentation
of the CAM EUL andSLD dynamicalcores(Collinset al 2004).CAM EUL and
SLDapplythefourth-ordertemperaturediffusion
"T "p
FT = K4 ’4T ps ’4lnps (13.13)
− ’ − "p"ps (
where pisthepressureand p symbolizesthesurfacepressure.Thesecondtermin
s
FT consistsoftheleadingterminthetransformationofthe’4operatorfrom*sur-
facestopressuresurfaces.CAMalsoappliesasecond-ordersponge-layerdiffusion
atupperlevels.ButsincetheupperlevelsinCAMcoincidewithpurepressurelevels
thecorrectionisnotneededthere.Ingeneral,itisunclearwhetherdiffusionshould
beappliedalongconstantmodellevels,constantpressurelevelsorevenalongcon-
stant height or isentropic levels. If the diffusion primarily counteracts numerical
artifacts,argumentscanbefoundthatitshouldbeappliedalongmodellevels.How-
ever,iftheprimarymotivationistocharacterizephysicalmixing,height,pressureor
isentropiclevelsareadvantageousasexplainedindetailbyStaniforthetal(2006).
NotethatNCAR’sEULandSLDdynamicalcoresactuallyapplyavariantofthe
diffusionshowninEqs.(13.10)and(13.11)totherelativevorticityandhorizontal
divergencefieldswhichgeneralizestheapproachbyBourkeetal(1977).Itisgiven
by
2 q
F$ =(−1)q+1K2q ’2q($+f)+(−1)q+1($+f) a2 (13.14)
’ ) * (
2 q
F% =(−1)q+1K2q ’2q%+(−1)q+1% a2 (13.15)
’ ) * (